Yes, a founder can be on the board. Should be on the board. The board is composed of members who represent the interests of the owner. If the owner or founder has the time, skills and networks it requires to win, s/he is the best person to be on the board for unprecedented success since they have lots of passion and goodwill than any other person!
Unlike public entities where there is need for separation of powers or independence of executive, board and founder, for not for profits (NFPs), the founder involvement is the best.
Brac, the leading global NFP with fantastic ecosystem and growth, had the founder as a board member until he was unable and the organization succeeded to the extent that it no longer needed him. So bad the world lost him late last year!
At the start, any business needs the owner involvement to grow. It is called sweet capital. The issue is, which role would give the business the best value -executive director, board member or chair or just founder who is not involved in day to day and board issues? It makes more sense to provide support as board member until you are happy the organization does not need you!